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Let $N$ and $K$ be groups and let $G$ be an extension of $N$ by $K$. Given a property $𝒫$ of group compactifications, one can ask whether there exist compactifications $N^{\text{'}}$ and $K^{\text{'}}$ of $N$ and $K$ such that the universal $𝒫$-compactification of $G$ is canonically isomorphic to an extension of $N^{\text{'}}$ by $K^{\text{'}}$. We prove a theorem which gives necessary and sufficient conditions for this to occur for general properties $𝒫$ and then apply this result to the almost periodic and weakly almost periodic compactifications of $G$.